FACTORISATION PROPERTIES OF GROUP SCHEME
ACTIONS
RUDOLF TANGE
Summary. Let H be an algebraic group scheme over a field k acting on
a commutative k-algebra A which is a unique factorisation domain. We
show that, under certain mild assumptions, the monoid of nonzero H-stable
principal ideals in A is free commutative. From this we deduce, in certain
special cases, results about the monoid of nonzero semi-invariants and the
algebra of invariants. We use an infinitesimal method which allows us to
work over an arbitrary base field.
1. Introduction and notation
Let H be a linear algebraic group acting on an irreducible variety X. It
is of interest to know conditions for when a function f on X is an H-(semi)invariant. For example, when X is normal, H is connected and the principal
divisor (f ) is fixed by a closed subgroup H ′ of H, then f is an H ′ -semi-invariant.
Without the action of the bigger connected group H this is no longer true.
Another well-known fact is that when H is connected, X is affine and k[X] is
a unique factorisation domain (UFD) of which the invertible elements are the
nonzero constants, then the prime factors of an H-semi-invariant are H-semiinvariants. For non-connected groups and/or invariants the situation is much
more complicated. A standard example where the ring of invariants in a UFD
is not a UFD is k[PGLn ] = k[GLn ]Z , where Z consists of the nonzero multiples
of the identity acting via the right regular representation. See [1, Ch. 3] for the
case of invariants for finite groups.
The quotient space X/H (when it exists) is often also involved here. By [4,
III §3 no. 2,5] every homogeneous space for a linear algebraic group G is of the
form G/H, where H is a closed subgroup scheme of G (so k[H] need not be
reduced). See [2, V.17] for a special case. So even if one is only interested in
homogeneous spaces (over fields of positive characteristic) one is led to consider
group scheme actions.
In this paper we study the behaviour of factorisation with respect to the
action of a group scheme H. Our initial interest was in (semi-)invariants, but it
turns out, for reasons partly indicated above, that we first have to look at the
property “Aa is H-stable” of an element a ∈ A. After that we indicate ways
to go from“Aa is H-stable” to “a is an H-semi-invariant” (see Remark 2.1 and
Proposition 2). The main results are Theorems 1 and 2. They state roughly
that when a group scheme H acts on a UFD A, then the monoid of nonzero
elements a ∈ A such that Aa is H-stable has the unique factorisation property.
2010 Mathematics Subject Classification. 13F15, 14L15, 14L30.
1
2
R. H. TANGE
Throughout this paper k denotes a field. All group schemes are affine and H
will always denote an algebraic group scheme over k. For the basic definitions
concerning (group) schemes we refer to [7]. By “algebraic group” we will always
mean a reduced linear algebraic group over an algebraically closed field. By
“reductive group” we will always mean an algebraic group with trivial unipotent
radical. An element of a module for a group scheme is called a semi-invariant
if it spans a submodule.
2. Group scheme actions on UFD’s
We start with a basic result about distributions. We need a simple lemma
which is easily proved using the modular property for subspaces of a vector
space and induction on r.
Lemma 1. Let V0 ⊇ V1 ⊇ V2 ⊇ · · · ⊇ Vr and W0 ⊇ W1 ⊇ W2 ⊇ · · · ⊇ Wr be
two descending chains of subspaces of a vector space V . Then
r
∩
(Vi + Wr−i ) = Vr + Wr +
i=0
r−1
∑
Vi ∩ Wr−1−i .
i=0
Now let
be affine algebraic k-schemes and let x ∈ X(k) and x′ ∈
As in [7] we denote the space of distributions of X with support in x by
Dist(X, x) ⊆ k[X]∗ . It consists of the functionals that vanish on some power of
the vanishing ideal Ix of x. This space has a natural filtration (Distn (X, x))n≥0 ,
where Distn (X, x) consists of the functionals that vanish on the (n+1)-st power
of the vanishing ideal Ix of x. If we apply Lemma 1 with r = n, Vi = Ixi+1 ⊗k[X ′ ]
and Wi = k[X] ⊗ Ixi+1
′ , then we obtain, as in [7, I.7.4],
X ′ (k).
n
∩
X, X ′
n
∑
n+1
n+1
′
(Ixi+1 ⊗k[X ′ ]+k[X]⊗Ixn+1−i
)
=
I
⊗k[X
]+k[X]⊗I
+
Ixi ⊗Ixn+1−i
(1)
′
′
x
x′
i=0
i=1
∑n+1
which is clearly equal to i=0 Ixi ⊗ Ixn+1−i
. From this it is deduced in [7, I.7.4]
′
that there is an isomorphism
Dist(X,
x)
⊗
Dist(X ′ , x′ ) ∼
= Dist(X × X ′ , (x, x′ ))
∑n
which maps each i=0 Disti (X, x) ⊗ Distn−i (X, x) onto Distn (X × X ′ , (x, x′ )).
Taking X = X ′ and x = x′ we see that Dist(X, x) is a coalgebra. Its comultiplication ∆ is the “differential” of the diagonal embedding at (x, x). The counit
is the evaluation at the constant function 1 ∈ k[G].
If we apply Lemma 1 with r = n − 2, Vi = Ixi+2 ⊗ k[X ′ ] and Wi = k[X] ⊗ Ixi+2
′ ,
then we obtain
n−1
∩
(Ixi+1 ⊗k[X ′ ]+k[X]⊗Ixn+1−i
)
′
=
Ixn ⊗k[X ′ ]+k[X]⊗Ixn′ +
i=1
n−1
∑
Ixi ⊗Ixn+1−i
. (2)
′
i=2
Lemma 2. Let X be an algebraic affine k-scheme, let x ∈ X(k), let n > 0 and
denote the evaluation at x by εx . Then we have for all u ∈ Distn (X, x) that
∆(u) − u ⊗ εx − εx ⊗ u ∈
n−1
∑
i=1
Disti (X, x) ⊗ Distn−i (X, x).
FACTORISATION PROPERTIES OF GROUP SCHEME ACTIONS
Proof. By (2)
∑n−1
i=1
3
Disti (X, x) ⊗ Distn−i (X, x) is everything that vanishes on
Ixn ⊗ k[X] + k[X] ⊗ Ixn +
n−1
∑
Ixi ⊗ Ixn+1−i .
i=2
From (1) and the fact that ∆ is filtration preserving we get ∆(u) vanishes on
∑n−1 i
n+1−i and, clearly, the same holds for u ⊗ ε and ε ⊗ u, so we
x
x
i=2 Ix ⊗ Ix
only have to check that ∆(u) − u ⊗ εx − εx ⊗ u vanishes on Ixn ⊗ k[X] and
k[X] ⊗ Ixn . This follows immediately from u(Ixn+1 ) = 0, ∆(u)(f ⊗ g) = u(f g)
and g = g − g(x) + g(x) for f, g ∈ k[X].
A simple induction gives us the following generalisation.
Corollary. Let X, x, n, εx be as in Lemma 2, let m ≥ 2 and let ∆m : k[X] →
k[X]⊗m be the m-th comultiplication. Then we have for all u ∈ Distn (X, x) that
∆m (u) − u ⊗ εx ⊗ · · · ⊗ εx − · · · − εx ⊗ · · · ⊗ εx ⊗ u
∑
Disti1 (X, x) ⊗ · · · ⊗ Distim (X, x),
∈
i1 ,...,im
where the sum is over all i1 , . . . , im in {0, . . . , n − 1} with
∑m
j=1 ij
= n.
We will apply the above lemma and its corollary to the case that X = H,
where H is an (affine) algebraic group scheme and x is the identity e ∈ H.
Then we write Dist(H, e) = Dist(H). Since H is a group scheme, Dist(H) is
not just a coalgebra, but a Hopf algebra. Its unit element it the evaluation ε at
e. We note that Dist(H) = Dist(H 0 ), where H 0 is the identity component of
H. So we can speak about Dist(H)-modules (because of the algebra structure)
and about the tensor product of Dist(H)-modules. Recall that H is called finite
when dim k[H] < ∞ and infinitesimal when k[H] is finite dimensional and has
a unique maximal ideal. Every H-module is a Dist(H)-module and if H is
infinitesimal, then this gives an equivalence of categories. In general a (left)
H-module ([7, I.2.7]) is the same thing as a right k[H]-comodule (for k[H] as a
coalgebra). If k is perfect of characteristic p > 0, r ≥ 0 and H is an algebraic
group scheme and over k, then the r-th Frobenius kernel ∪
Hr of H (see [7, I.9])
is an infinitesimal group scheme and we have Dist(H) = r≥0 Dist(Hr ).
Let A be a k-algebra. We say that A is a Dist(H)-algebra, if it is a Dist(H)module such that the multiplication A ⊗ A → A is a morphism of Dist(H)modules. We say that A is an H-algebra or that H acts on A, if A is an
H-module such that, for each commutative k-algebra R, H(R) acts by automorphisms of the algebra R ⊗ A. This condition is equivalent to the condition
that the comodule map ∆A : A → A ⊗ k[H] is a homomorphism of algebras.
Every H-algebra is a Dist(H)-algebra and when H is infinitesimal, the two
notions coincide.
Let A be a commutative Dist(H)-algebra. Assume first that k is perfect with
char k = p > 0 and that H is infinitesimal of height ≤ r (i.e. H = Hr ). Then
r
pr
pr
pr
we have for ∑
f ∈ k[H] that f p = (f
∑ − f (e)) + f (e) = f (e) . Furthermore,
if ∆A (a) = i ai ⊗ fi , then a = i fi (e)ai . Finally, A is an H-algebra, since
H is infinitesimal. So the comodule map is a homomorphism of algebras. From
4
R. H. TANGE
r
these facts we easily deduce that H fixes the elements of Ap . Now assume only
that char k = p > 0. Then we deduce from the above, by base extension to a
r
perfect field, that u ∈ Dist(H) acts Ap -linearly on A whenever it kills the pr -th
powers of the elements of Ie .
Now drop the assumption on k and assume that A is a domain. Then
the Dist(H)-action extends to give Frac(A) the structure of a Dist(H)-algebra
(clearly such an extension is unique). To see this in case char k = 0 one can
use Cartier’s Theorem ([4, II.6.1.1]) which says that Dist(H) = U (Lie(H)).
Then the action of Dist(H) is given by the standard extension of derivations
to the field of fractions. In case char k = p > 0 we use that, by the above, for
r
n ≤ pr − 1, the elements of Distn (H) act Ap -linearly on A. So one can extend
r
their action to Frac(A) = A[(Ap \ {0})−1 ] in the obvious way. This clearly
leads to the required action of Dist(H). If A is an H-algebra, then there is,
of course, also an action of H(k) on Frac(A). It is important to note that the
actions of Dist(H) and H(k) on Frac(A) are in general not locally finite.
If A is a Dist(H)-algebra and a ∈ A, then Aa is Dist(H)-stable if and only if
u · a ∈ Aa for all u ∈ Dist(H). If, in addition, A is a commutative domain, then
we also have for a ∈ Frac(A) that Aa is Dist(H)-stable if and only if u · a ∈ Aa
for all u ∈ Dist(H). We note that if a group G acts on a commutative domain
A by automorphisms, then Aa is G-stable if and only if, for all g ∈ G, a and
g(a) differ by a unit.
The first assertion of the lemma below is [7, I.7.17(6), 8.6]. To prove the
second assertion one may assume that k1 = k. Then one takes a k-point from
each irreducible component of Hk and the result follows from [7, I.7.17(6), 8.6]
and some elementary properties of the comodule map. We leave the details to
the reader.
Lemma 3. Let H be an algebraic group scheme over k and let M be an Hmodule. Assume that k is perfect or that H is finite and let N be a subspace of
M . Denote the identity component of H by H 0 .
(i) N is an H 0 -submodule if and only if N is Dist(H)-stable.
(ii) If k1 is an extension field of k such that every irreducible component of
Hk1 contains a k1 -point, then N is an H-submodule if and only if N is
Dist(H)-stable and k1 ⊗ N is H(k1 )-stable.
Proposition 1. Let H be an algebraic group scheme over k and let A be a
Dist(H)-algebra which is a commutative domain. Let a, b, c ∈ A and m ≥ 0.
(i) Assume that A is a UFD and that b and c are coprime. If Abc is Dist(H)stable, then so are Ab and Ac.
(ii) Assume that A is a UFD and that b and c are coprime. If A(b/c) is
Dist(H)-stable, then so are Ab and Ac.
(iii) If Aam is a Dist(H)-stable and m ̸= 0 in k, then Aa is Dist(H)-stable.
Proof. (i). We show by induction on n that Ab is Distn (H)-stable. For n = 0
there is nothing to prove. So assume that n > 0. Let u ∈ Distn (H). By
∑n−1 1
Lemma 2 we can write ∆(u) − u ⊗ ε − ε ⊗ u = i=1
ui ⊗ u2n−i , where ε is the
j
evaluation at the unit element of H and ui ∈ Disti (H). Since A is a Dist(H)∑n−1 1
(ui · b)(u2n−i · c). So by our
algebra we have u · (bc) − (u · b)c − b(u · c) = i=1
FACTORISATION PROPERTIES OF GROUP SCHEME ACTIONS
5
assumption and the induction hypothesis we get that b divides (u · b)c. Since b
and c are coprime, this means that b divides (u · b).
(ii). Let u ∈ Distn (H). Since b = cb c, we have by Lemma 2
∑
b
b
b
u · b − (u · ) c − (u · c) =
(u1i · )(u2n−i · c).
c
c
c
n−1
i=1
So we have in A that
∑
b
b
(u · b) c − (u · ) c2 − b(u · c) =
(u1i · ) c (u2n−i · c).
c
c
n−1
i=1
Now, by assumption, (u ·
∈ Ab and
∈ Ab, so b divides (u · b) c. And
therefore b divides u · b. Furthermore we obtain, using induction as in (i), that
c divides b(u · c) and therefore that c divides u · c.
(iii). Let u ∈ Distn (H). By the corollary to Lemma 2 we can write
∑
u1j ⊗ · · · ⊗ um
∆m (u) = u ⊗ ε ⊗ · · · ⊗ ε + · · · + ε ⊗ · · · ⊗ ε ⊗ u +
j ,
b
c) c
(u1i · cb ) c
j
where
Then
uij
∈ Distnij (H) for nij ∈ {0, . . . , n − 1} with
u · (am ) = mam−1 u · a +
∑
∑m
i=1 nij
= n for all j.
u1j · a · · · um
j · a.
j
So, by induction on n, we get that a divides u · a for all u ∈ Distn (H).
Corollary. Assume that A is an H-algebra and that k is perfect or H is finite.
(i) If H is irreducible, then, in the conclusions in Proposition 1, “Dist(H)stable” may be replaced by “H-stable”.
(ii) Assume that A is normal and that every irreducible component of H contains a k-point. Then Proposition 1(iii) is also valid with “Dist(H)-stable”
replaced by “H-stable”. Furthermore, if in (ii) we require A(b/c) also to
be H(k)-stable, then Ab and Ac are H-stable.
Proof. (i). This follows immediately from [7, I.7.17(6), 8.6].
(ii). Let h ∈ H(k). First consider Proposition 1(ii). Then, by assumption,
(h · b)c ∈ Ab(h · c) and (h−1 · b)c ∈ Ab(h−1 · c). So b divides h · b and c divides
h · c. Now the result follows from Lemma 3. Now consider Proposition 1(iii).
Let h ∈ H(k). By assumption we have am |(h · a)m . Then (h · a)/a ∈ Frac(A)
is integral over A. So a|h · a, since A is normal. Now the result follows again
from Lemma 3.
Remark 1. The arguments in the proof of Proposition 1(iii) also yield the
statement with the property “Aa is Dist(H)-stable” replaced by “a is Dist(H)semi-invariant” or by “a is Dist(H)-invariant”.
The theorem below is a generalisation of a well-known result for connected
algebraic groups over an algebraically closed field (see e.g. [6, Lem. 20.1]). We
remind the reader that the unique factorisation property makes sense for any
commutative monoid in which the cancellation law holds. In fact such a monoid
6
R. H. TANGE
has the unique factorisation property if and only if the quotient by the units is
a free commutative monoid.
Theorem 1. Let H be an irreducible algebraic group scheme over k and let A
be a Dist(H)-algebra. Assume that A is a UFD. Then the monoid of nonzero
elements a ∈ A such that Aa is Dist(H)-stable has the unique factorisation
property. If char k = 0, then its irreducible elements are the irreducible elements
a of A such that Aa is Lie(H)-stable. If char k = p > 0, then its irreducible
s
s
elements are the elements ap , where a is irreducible in A, s ≥ 0, Aap is
Dist(H)-stable, and s is minimal with this property.
Proof. The case that char k = 0 follows easily from Cartier’s Theorem and
Proposition 1, so we assume that char k = p > 0. Then we note that the
statement of the theorem is equivalent to the following statement. If a is an
irreducible factor which occurs to the power t in the prime factorization in A of
s
an element b ∈ A such that Ab is Dist(H)-stable, then Aap is Dist(H)-stable
for some s ≥ 0 and if s is minimal with this property, then ps |t.
So let a, b, t be as stated above. Then, by Proposition 1(i), Aat is Dist(H)r
r
stable. Now write t = pr t1 with p - t1 . Then at = (ap )t1 , so Aap is Dist(H)stable by Proposition 1(iii). For s as in the theorem we must have s ≤ r and
therefore ps |t.
To formulate the next theorem correctly we need some notation. Let H be
an algebraic group scheme over k and let A be an H-algebra which is a UFD.
Let M be the quotient of the monoid of nonzero elements in A such that Aa is
H 0 -stable by the units. So M can also be considered as the monoid of nonzero
H 0 -stable principal ideals in A. Furthermore, we put Γ = H(k)/H 0 (k). Note
that Γ is a finite group and that it acts on M .
Theorem 2. Let H be an algebraic group scheme over k and let A be an
H-algebra. Assume that A is a UFD, that every irreducible component of H
contains a k-point and that k is perfect or H is finite. Let M and Γ be as
above. Then the monoid of nonzero elements a ∈ A such that Aa is H-stable
has the unique factorisation property. Its irreducible elements are the products
of representants of the elements in the Γ-orbits of the irreducible elements of
M.
Proof. By Theorem 1 and Lemma 3(i) M is a free commutative monoid. By
Lemma 3(ii), an element Aa of M is H-stable if and only if it is Γ-fixed. So
the quotient of the monoid from the theorem by the units is M Γ . But this
monoid is free commutative with its irreducible elements as described in the
theorem.
Remarks 2. 1. If A has a Dist(H)-stable filtration A0 ⊆ A1 ⊆ A2 · · · with
A0 = k such that grA is a domain, then Aa is Dist(H)-stable if and only if a
is a Dist(H)-semi-invariant. This follows from a simple degree comparison. By
Lemma 3 we can draw these conclusions for an H-action, if we assume that k
is perfect or H is finite.
2. Assume that k is perfect with char k = p > 0. Let H be an algebraic
group scheme over k and let A be a reduced commutative H-algebra. Let
FACTORISATION PROPERTIES OF GROUP SCHEME ACTIONS
7
A(r) and H (r) be the r-th Frobenius twists of A and H, see [7, I.9.2]. Then
r
r
H (r) acts on A(r) and for the isomorphism a 7→ ap : A(r) → Ap we have
r
Frr (h) · a = h · ap for all a ∈ A, all h ∈ H(R) and all commutative k-algebras
R. Here Frr : H → H (r) is the r-th Frobenius morphism. By [7, I.9.5] we have
r
r
k[H/Hr ] = k[H]Hr = k[H]p . So, if for the nil radical n of k[H] we have np = 0,
∼
(r)
then Frr induces an isomorphism H/Hr → Hred , where Hred is the closed subgroup scheme of H defined by n. So a ∈ A is an Hred -(semi-)invariant if and
r
(r)
only if it is an Hred -(semi-)invariant as an element of A(r) if and only if ap
is an H-(semi-)invariant. Similarly, we get that Aa is Hred -stable if and only
r
r
if Ap ap is H-stable. If A is a normal domain, then this is also equivalent to
r
Aap is H-stable.
3. Let k and H be as in the previous remark and assume that H is reduced. Let
A be a reduced commutative Dist(H)-algebra. Then, by [7, I.9.5], Frr induces
∼
(r)
an isomorphism Hr+s /Hr → Hs . So, as in the previous remark, we obtain that
r
a ∈ A is an Hs -(semi-)invariant if and only if ap is an Hr+s -(semi-)invariant. So
s
the minimal s such that ap is an Hr -(semi-)invariant is r−t where t ∈ {0, . . . , r}
is maximal with the property that a is an Ht -(semi-)invariant. The analogues
of the statements from the previous remark about the property that Aa is Hstable are also valid.
4. Under the assumptions of Theorem 1 we also have that the group of nonzero
Dist(H)-stable principal fractional ideals is free abelian. In the case of Theorem 1 the group of nonzero principal fractional ideals that are Dist(H)-stable
and H(k)-stable is free abelian. The extra argument one needs is Proposition 1(ii) and assertion (ii) of its corollary.
5. The UFD property is badly behaved with respect to field change, see [3,
Chap. 7 Ex. 4 and 6 to §3].
By Remark 2.1 filtrations give us an obvious way to deduce that a is an Hsemi-invariant from the fact that Aa is H-stable. It applies, for example, when
A = k[V ], where V is a finite dimensional H-module. In Proposition 2 below
we replace the filtration by the action of a “big” (AG = k) reductive group G.
It applies, for example, to the case A = k[G] and H ≤ G acting via the right
or left regular action. It also applies to the case A = k[G] and H ≤ G ≤ G × G
where G is embedded diagonally and G × G acts in the usual way (so G × G is
now the big reductive group and H acts via the adjoint action). In both cases
we take H ′ = 1.
We emphasize that the property that every a ∈ A such that Aa is H-stable is
an H-semi-invariant is a very general one. For example, when H is a (reduced)
closed subgroup of a connected algebraic group acting on an affine variety X,
then the algebra k[X] has this property. This follows from the result stated at
the beginning of Section 1 by passing to the normalisation of X, since, when X
is normal, k[X]f is H-stable if and only if (f ) is H-fixed. So Remark 2.1 and
Proposition 2 below are not the definitive results in this direction. The group of
order two acting on the one-dimensional torus by inversion is an example where
k[X] does not have this property. To prove Proposition 2 we need a lemma.
8
R. H. TANGE
Lemma 4. Assume that k is algebraically closed. Let T be a torus acting on a
commutative domain A and let a, b, c ∈ A \ {0}. Assume that a = bc and that
for each weight χ of T the χ-component of c is nonzero if the χ-component of
a is nonzero. Then b is T -fixed.
Proof. The assumption on the weight components is inherited by any subtorus
of T , so we may assume that T is one-dimensional. Then the T -action amounts
to a Z-grading of the algebra A and the result follows by comparing highest
and lowest degree.
Proposition 2. Assume that k is algebraically closed. Let G and H ′ be algebraic group schemes over k acting on a k-algebra A and assume that these
actions commute. Assume furthermore that G is a connected reductive algebraic
group and that A is a commutative domain with AG = k. Let H be a closed
subgroup scheme of G and let a ∈ A. If Aa is stable under H and H ′ , then a
is a semi-invariant for H and H ′ .
Proof. Let A0 ⊆ A1 ⊆ A2 ⊆ · · · be the filtration of A from [6, §15] (first
introduced in [10, §4]). Then this filtration is G and H ′ -stable, gr(A) is a
domain and A0 is spanned by the G-semi-invariants. Taking degrees we get
that for h ∈ H(k) and u ∈ Dist(H) we have h · a ∈ A0 a and u · a ∈ A0 a, and
the same with H replaced by H ′ . Since characters of G are determined by their
restriction to the connected centre, the result follows from Lemma’s 4 and 3
and the fact that AG = k.
We note that, although Proposition 2 is only stated for k algebraically closed,
one can in certain situations apply it to the case that k is not algebraically closed
by applying field extension from k to k. The point is that the properties “Aa
is H-stable” and “a is an H-semi-invariant” are well-behaved with respect to
field extension and restriction. To make this work one needs, of course, that
k ⊗ A is a domain.
The following result is an immediate consequence of Theorem 2, Proposition 2
and the well-known fact that, for G connected reductive, k[G] is a UFD if its
derived group is simply connected (see [12] for references and for an elementary
proof).
Corollary 1. Assume that k is algebraically closed. Let G be a connected
reductive algebraic group over k with simply connected derived group and let
H be a closed subgroup scheme of G. Then the monoid of nonzero H-semiinvariants in k[G] under the right regular action has the unique factorisation
property. Its irreducible elements are as given by Theorem 2 with the property
“Aa is H-stable” replaced by “a is an H-semi-invariant”.
We finish the paper with some generalisations of results in [12] and [13]. For
results on generators of algebras of infinitesimal invariant we refer to [5]. We
need some terminology. Assume that k is algebraically closed and let G be a
connected reductive group over k. As in [9] we call an element ξ of g∗ semisimple if there exists a maximal torus T of G such that ξ vanishes on all the
root spaces of g relative to T .
FACTORISATION PROPERTIES OF GROUP SCHEME ACTIONS
9
Corollary 2. Assume that k is perfect of characteristic p > 0. Let G be an
algebraic group scheme over k such that Gk is a connected reductive algebraic
group and let g be its Lie algebra. In the following cases AGr is a UFD and
r−t
its irreducible elements are the elements ap , where a is irreducible in A and
t ∈ {0, . . . , r} is maximal with the property that a is an Ht -invariant.
(1) If A = k[G] is a UFD,
(2) If A = k[g] and the semi-simple elements are dense in Lie(Gk ) = k ⊗ g,
(3) If A = S(g) = k[g∗ ] and the semi-simple elements are dense in (k ⊗ g)∗ ,
where in each case Gr ≤ G acts via the adjoint action.
Proof. By Theorem 1 and Remark 2.3 we only have to check that in each case
a ∈ A is a Gr -invariant whenever Aa is Dist(Gr )-stable. For this we may assume
that k is algebraically closed. By Proposition 2 and Remark 2.1 it remains to
show that every semi-invariant of Gr in A is an invariant. Assume, in case
(1), that f ∈ k[G] is a nonzero semi-invariant for Gr . By the density of the
semi-simple elements in G there exists a maximal torus T such that f |T ̸= 0.
Now we obtain, using the triviality of the adjoint action of T on k[T ] as in [12,
Lem. 2], that f is a Dist(T )-invariant. On the other hand, f is also an invariant
of the Dist(Uα,r ), since the infinitesimal root subgroups are unipotent. So, e.g.
by [7, II.3.2], f is a Dist(Gr )-invariant, that is, a Gr -invariant. In the other two
cases the proof is completely analogous.
For the next corollary we need to assume the so-called “standard hypotheses”
for a reductive group in positive characteristic. They can be found in [8] and
are also stated in the introduction of [13].
Corollary 3. Assume that k is algebraically closed of characteristic p > 0. Let
G be a connected reductive algebraic group over k, let g be its Lie algebra, let U
be the universal enveloping algebra of g and let r ≥ 1. Assume that G satisfies
the standard hypotheses (H1)-(H3) from [8]. Then U Gr is a UFD.
Proof. We have U Gr = Z Gr , where Z = U G1 = U g is the centre of U . By [13,
Thm. 2] Z is a UFD and by [13, 1.4] Z has a G-stable filtration A0 ⊆ A1 ⊆
A2 ⊆ · · · with A0 = k such that gr(Z) is a domain. So, by Remark 2.1, we
have for a ∈ Z that Za is Gr -stable if and only if a is a Gr -semi-invariant.
Furthermore, Z ∼
= S(g)g ⊆ S(g) as G-modules by [13, 1.4], so by (iii) of the
preceding corollary, every Gr -semi-invariant of Z is a Gr -invariant. Now the
result follows from Theorem 1.
Remarks 3. 1. If k is algebraically closed, or more generally G is split over k,
then k[G] is a UFD if and only if the derived group DG is simply connected.
The arguments in [12, Sect. 1] are also valid in the more general case that G is
split over k.
2. If k = R and G = SO2 is the compact real form of the one dimensional
complex torus (see [2, II.8.16]), then R[G] ∼
= R[a, b]/(a2 + b2 − 1) is not a UFD.
See also [3, Chap. 7 Ex. 4 to §3]. The same is true for SO2 over any field in
which −1 is not a square.
3. The following example shows that the assumption on the density of the
semi-simple elements in case (3) of Corollary 2 cannot be omitted. Let k be
10
R. H. TANGE
algebraically closed of characteristic 2. Then pgl∗2 ∼
= sl2 as GL2 -modules. So
∗
the semi-simple elements are not dense in pgl2 . Put G = PGL2 and let (h, x, y)
be a basis of pgl2 with [h, x] = x, [h, y] = y and [x, y] = 0. Then it follows from
the identity (xy)p = xp y p that S(g)G1 = S(pgl2 )pgl2 = k[xp , y p , hp , xy] is not a
UFD. Note that x and y are pgl2 -semi-invariants that are not pgl2 -invariants.
Acknowledgement. This research was funded by a research grant from The
Leverhulme Trust.
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Department of Mathematics, University of York, Heslington, York,
UK, YO10 5DD. E-mail address : [email protected]